KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_textbook_ex4_bb3_in

Found invariant 1 ≤ X₁ for location eval_textbook_ex4_bb1_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_textbook_ex4_bb4_in

Found invariant 1+X₃ ≤ X₁ ∧ 1 ≤ X₁ for location eval_textbook_ex4_bb5_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_textbook_ex4_bb2_in

Found invariant 1+X₃ ≤ X₁ ∧ 1 ≤ X₁ for location eval_textbook_ex4_stop

Problem after Preprocessing

Start: eval_textbook_ex4_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_textbook_ex4_bb0_in, eval_textbook_ex4_bb1_in, eval_textbook_ex4_bb2_in, eval_textbook_ex4_bb3_in, eval_textbook_ex4_bb4_in, eval_textbook_ex4_bb5_in, eval_textbook_ex4_start, eval_textbook_ex4_stop
Transitions:
t₁: eval_textbook_ex4_bb0_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb1_in(X₀, 1, X₂, X₃)
t₂: eval_textbook_ex4_bb1_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb2_in(1, X₁, X₂, X₃) :|: X₁ ≤ X₃ ∧ 1 ≤ X₁
t₃: eval_textbook_ex4_bb1_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb5_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ X₁ ∧ 1 ≤ X₁
t₄: eval_textbook_ex4_bb2_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃
t₅: eval_textbook_ex4_bb2_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃
t₆: eval_textbook_ex4_bb3_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb2_in(1+X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₇: eval_textbook_ex4_bb4_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb1_in(X₀, 1+X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃
t₈: eval_textbook_ex4_bb5_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_stop(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ 1+X₃ ≤ X₁
t₀: eval_textbook_ex4_start(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂: eval_textbook_ex4_bb1_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb2_in(1, X₁, X₂, X₃) :|: X₁ ≤ X₃ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₃+2 {O(n)}

MPRF:

• eval_textbook_ex4_bb1_in: [1+X₃-X₁]
• eval_textbook_ex4_bb2_in: [X₃-X₁]
• eval_textbook_ex4_bb3_in: [X₃-X₁]
• eval_textbook_ex4_bb4_in: [X₃-X₁]

MPRF for transition t₅: eval_textbook_ex4_bb2_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃+2 {O(n)}

MPRF:

• eval_textbook_ex4_bb1_in: [1+X₃-X₁]
• eval_textbook_ex4_bb2_in: [1+X₃-X₁]
• eval_textbook_ex4_bb3_in: [1+X₃-X₁]
• eval_textbook_ex4_bb4_in: [X₃-X₁]

MPRF for transition t₇: eval_textbook_ex4_bb4_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb1_in(X₀, 1+X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

2⋅X₃+1 {O(n)}

MPRF:

• eval_textbook_ex4_bb1_in: [2⋅X₃-X₁]
• eval_textbook_ex4_bb2_in: [2⋅X₃-X₁]
• eval_textbook_ex4_bb3_in: [2⋅X₃-X₁]
• eval_textbook_ex4_bb4_in: [2⋅X₃-X₁]

MPRF for transition t₄: eval_textbook_ex4_bb2_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

2⋅X₂⋅X₃+2⋅X₂ {O(n^2)}

MPRF:

• eval_textbook_ex4_bb1_in: [X₂]
• eval_textbook_ex4_bb2_in: [1+X₂-X₀]
• eval_textbook_ex4_bb3_in: [X₂-X₀]
• eval_textbook_ex4_bb4_in: [X₂-X₀]

MPRF for transition t₆: eval_textbook_ex4_bb3_in(X₀, X₁, X₂, X₃) → eval_textbook_ex4_bb2_in(1+X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

2⋅X₂⋅X₃+2⋅X₂ {O(n^2)}

MPRF:

• eval_textbook_ex4_bb1_in: [X₂]
• eval_textbook_ex4_bb2_in: [1+X₂-X₀]
• eval_textbook_ex4_bb3_in: [1+X₂-X₀]
• eval_textbook_ex4_bb4_in: [X₂-X₀]

Found invariant 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_textbook_ex4_bb3_in_v2

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_textbook_ex4_bb4_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ for location eval_textbook_ex4_bb2_in

Found invariant 1+X₃ ≤ X₁ ∧ 1 ≤ X₁ for location eval_textbook_ex4_stop

Found invariant 1 ≤ X₁ for location eval_textbook_ex4_bb1_in

Found invariant 1+X₃ ≤ X₁ ∧ 1 ≤ X₁ for location eval_textbook_ex4_bb5_in

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ for location eval_textbook_ex4_bb3_in_v1

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_textbook_ex4_bb2_in_v1

All Bounds

Timebounds

Overall timebound:4⋅X₂⋅X₃+4⋅X₂+4⋅X₃+9 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+2 {O(n)}
t₃: 1 {O(1)}
t₄: 2⋅X₂⋅X₃+2⋅X₂ {O(n^2)}
t₅: X₃+2 {O(n)}
t₆: 2⋅X₂⋅X₃+2⋅X₂ {O(n^2)}
t₇: 2⋅X₃+1 {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₂⋅X₃+4⋅X₂+4⋅X₃+9 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+2 {O(n)}
t₃: 1 {O(1)}
t₄: 2⋅X₂⋅X₃+2⋅X₂ {O(n^2)}
t₅: X₃+2 {O(n)}
t₆: 2⋅X₂⋅X₃+2⋅X₂ {O(n^2)}
t₇: 2⋅X₃+1 {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 1 {O(1)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: 1 {O(1)}
t₂, X₁: 2⋅X₃+2 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀+X₂+5 {O(n)}
t₃, X₁: 2⋅X₃+3 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: X₂+4 {O(n)}
t₄, X₁: 2⋅X₃+2 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₂+5 {O(n)}
t₅, X₁: 2⋅X₃+2 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₂+4 {O(n)}
t₆, X₁: 2⋅X₃+2 {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: X₂+5 {O(n)}
t₇, X₁: 2⋅X₃+2 {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: X₀+X₂+5 {O(n)}
t₈, X₁: 2⋅X₃+3 {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}